vii
TABLE OF CONTENTS
CHAPTER
TITLE
PAGE
THESIS STATUS DECLARATION
SUPERVISOR’S DECLARATION
1
TITLE PAGE
i
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENTS
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
xiii
LIST OF FIGURES
xvi
LIST OF SYMBOLS AND ABREVIATIONS
xxiv
LIST OF APPENDICES
xxix
INTRODUCTION
1
1.1
Introduction
1
1.2
Background of the Research
6
1.3
Objective and Scope of Study
10
1.4
The Significance of this Research
11
1.5
Research Framework
12
1.6
Outline of Presentation
14
viii
2
MAGNETOENCEPHALOGRAPHY
17
2.1
Introduction
17
2.2
Electric Neural Activity
18
2.3
Current Dipole Model
22
2.4
Pattern Generated by a Current Dipole
24
2.5
Localization of Single Current Dipole
25
2.6
Detection of Magnetic Fields
26
2.6.1
The Forward Problem
27
2.6.2
The Backward Problem
28
2.7
Two Classes of Signals
28
2.8
Fuzzy Topographic Topological Mapping
32
Version 1 (FTTM1)
3
MATHEMATICAL BACKGROUND
38
3.1
Introduction
38
3.2
Magnetism
38
3.2.1
Magnetic Field Intensity
43
3.2.2
Material Permeability
43
3.2.2.1
Biot-Savart’s Law
43
The Magnetic Fields Effects
47
3.2.3
3.3
3.4
3.5
Crisp Set
49
3.3.1
50
Crisp Set Operations
Fuzzy Set
54
3.4.1
57
Fuzzy Set Operations
Image Processing
60
3.5.1
Neighbors of a Pixel
63
3.5.2
Connectivity
65
ix
4
3.5.3
Regions
66
3.5.4
Segmentation
66
3.5.5
Image Enhancement
67
3.6
Feature Extraction
67
3.7
Region Growing
67
3.8
Clustering
68
3.8.1
Fuzzy c-Means (FCM) Clustering
69
3.8.2
FCM Algorithm
70
3.8.3
Cluster Validity
74
MATHEMATICAL FORMULATION
78
4.1
Introduction
78
4.2
The Axis Convention
78
4.3
Coordinates Transformation
79
4.4
Two-Dimensional Representation of the
82
Magnetic Fields
4.5
The Measurement of Magnetic Field Lines
88
4.6
Mathematical Formulation
91
4.7
Acquisition of the Magnetic Fields Data
95
4.7.1
Generating the Magnetic Field Lines
97
4.7.1.1
97
Single Current Source
(Segment)
4.7.1.2
Double Current Sources
(Segments)
98
x
4.8
The Contour Plots of the Magnetic Fields for
98
Single Current Source
4.8.1
Approximate Current Location
101
4.8.2
The Direction of Current Flow
102
4.8.3
The Orientation of Current
102
4.9
The Contour Plots of Multiple Current Sources
103
4.10
Constraints in Generating the Magnetic
107
Fields Data
4.11
The Characteristics of Magnetic Fields Data on
107
the Measurement Plane
4.11.1 The Magnetic Fields for a Single
107
Current Source
4.11.2 Magnetic Fields for Double Current
108
Sources
5
THE FUZZY TOPOGRAPHIC TOPOLOGICAL
117
MAPPING VERSION 2 (FTTM2) ALGORITHM
5.1
Introduction
117
5.2
Fuzzy Topographic Topological Mapping
117
Version 2 (FTTM2)
5.3
Image Contour Plane
123
5.3.1
Data Pre-Processing
124
5.3.1.1
124
Data Conversion to their
Absolute Values
5.3.1.2
The Transformation of
Magnetic Fields Data to
Image Data
125
xi
5.3.2
The Application of Fuzzy c-Means
126
(FCM) Algorithm
5.3.3
The Application of Seed-Based Region
128
Growing (SBRG) Algorithm
5.4
Base Image Plane
129
5.5
Fuzzy Image Field
130
5.6
Topographic Image Field
131
5.6.1
Defuzzification of μ I z
131
5.6.2
View of Z in Three-Dimensions
133
5.6.3
The Inclination of Current Segments
133
5.6.4
Magnitude of Current Flow
133
5.6.4.1
133
Magnitude of Current for a
Single and Bounded Current
Segment
5.6.4.2
Magnitude of Current for
134
Multiple Bounded Current
Segments
5.7
6
Algorithm in FTTM2
FCM AND SBRG ON FTTM2: THEORETICAL
134
139
EMBEDDING
7
6.1
Introduction
139
6.2
Topology Preliminaries
141
6.3
FCM and SBRG on Topological Spaces
142
6.4
FCM and SBRG on FTTM2
144
IMPLEMENTATION
147
7.1
Introduction
147
7.2
The Implementation Process
154
xii
7.2.1
The Generated Data
154
7.2.1.1
154
Measurement on the Top of
the Head, B z
7.2.1.2
Measurement on the Left
194
Side of the Head, B y (left )
7.2.1.3
Measurement on the Right
205
Side of the Head, B y (right )
7.2.2
The Experimental Data
212
7.2.2.1
Experimental Set-up
214
7.2.2.2
Measuring The Experimental
216
Data
7.2.2.3
Limitations of the
230
Experimental Works
7.3
8
Results for other Datasets
232
7.3.1
Results from the Generated Data
232
7.3.2
Results from the Experimental Data
234
CONCLUSION
236
8.1
Summary of Research
236
8.2
Significant Contributions
238
8.3
Suggestions for Future Research
238
REFERENCES
241
Appendices A to
252-
E
271
xiii
LIST OF TABLES
TABLE NO.
TITLE
PAGE
7.1
The implementation table
149
7.2
The magnetic fields data, B z × 10
on the top of the head
7.3
The absolute values of the magnetic fields data from
Table 7.2
157
7.4
The image data
158
7.5
The matrix of features of size 3 by 256 for Feature
Type 1
161
7.6
The matrix of features of size 3 by 196 for Feature
Type 2
164
7.7
The matrix of features of size 7 by 196 for Feature
Type 3
168
7.8
The cluster validity measures
181
7.9
The sample of the fuzzified data, μ I
183
7.10
The defuzzified data, Z
184
7.11
The table of max(Z) from columns 1 to 11 of Table
7.10
188
7.12
The angles of inclination for the current segment for
measurement of the magnetic fields made on the xyplane (the top of the head)
191
7.13
The calculated current magnitude
192
5
(Tesla) measured
154
xiv
7.14
The magnetic fields data, B y (left ) measured on the left
side of the head
194
7.15
The image data for measurements made on the left
side of the head
197
7.16
The defuzzified data, Z
202
7.17(a)
The angles of inclination for the current segment 1
(S1)
204
7.17(b)
The angles of inclination for the current segment 2
(S2)
205
7.18
The magnetic fields data, B y (right ) measured on the
206
right side of the head
7.19
The image data
208
7.20(a)
Sample data of magnetic fields Bc measured in the
presence of current flowing
217
7.20(b)
Sample data of magnetic fields B0 measured in the
absence of current flowing
218
7.20(c)
Sample of the net magnetic fields, Bn = Bc − B0
219
7.20(d)
The magnetic field data where each data is multiplied
by 10 −7
220
7.21
The image data
223
7.22
The defuzzified data, Z
225
7.23
The table of max(Z) from columns 1 to 9 of Table 7.22
226
7.24
The angles of inclination for the current segment
where the measurement of the magnetic fields are
made on the xz-plane (the top of the head)
229
xv
7.25
The magnitude of current
231
7.26(a)
The results for current segment 1 (S1)
233
7.26(b)
The results for current segment 2 (S2)
233
7.27
The results for the angles of inclination
234
7.28
The angles of inclination
235
xvi
LIST OF FIGURES
FIGURE
TITLE
PAGE
NO.
1.1
The important parts of the human brain (left side)
2
1.2
(a) Schematic illustration of a pyramidal neuron and
three magnified synapses, (b) pyramidal neuron
3
1.3
Detection of cerebral magnetic fields using a SQUID
magnetometer
6
1.4
The forward neuromagnetic modeling
8
1.5
The inverse neuromagnetic modeling
8
1.6
Inverse neuromagnetic modeling: FTTM1
9
1.7
Fuzzy Topographic Topological Mapping Version 1
(FTTM1)
10
1.8
Research framework
13
1.9
Thesis outline
16
2.1
Schematic representations of the intracellular and
extracellular currents associated with the depolarization
front of the action potential
19
2.2
A schematic picture of the electrical currents during (a)
a propagating action potential, (b) a terminating action
potential, and (c) a postsynaptic potential
20
2.3
The orientation of the pyramidal cells in the cortex
21
2.4
Magnetic fields of a current dipole in a homogeneous
conducting medium
23
xvii
2.5
The contour plot of a current dipole
24
2.6
An example of the topographic field map calculated
from the measured MEG signals
25
2.7
The magnetic field readings of environmental noise and
human organs
27
2.8
Fuzzy Topographic Topological Mapping Version 1
(FTTM1)
34
3.1
The magnetic field lines around a magnet
39
3.2
The magnetic field lines surrounding a conductor
40
3.3
The direction of the magnetic fields lines using the righthand rule
41
3.4
The direction of the magnetic fields for current flowing
“into” the page
42
3.5
The direction of the magnetic fields for current that
emerges “out of” the page
42
3.6
The current segment AC
44
3.7
Two currents flowing in the opposite directions
48
3.8
Two currents flowing in the same direction
49
3.9
Operations on crisp sets
52
3.10
Membership function of fuzzy set
56
3.11
Fuzzy number F2 = {x ∈ ℜ : x is about 4}
57
3.12
Operations on fuzzy sets
59
3.13
Fundamental steps in digital image processing
62
3.14
A 3× 3 neighborhood about a point ( x , y ) in an image
64
xviii
3.15
The four neighborhood of ( x , y ) (4-neighbors)
64
3.16
Types of connectivity (a) 4-connectivity NSEW, (b) 6connectivity NW/SE, (c) 6-connectivity NE/SW, (d) 8connectivity
65
4.1
The axis convention
79
4.2
The cylindrical and the Cartesian coordinates
81
4.3
The transverse section of the current segment
83
4.4
The plot of B z against y
84
4.5
The relationship between the components in the
Cartesian coordinate system and cylindrical coordinate
system
85
4.6
The relationship between r and its y and z-components
86
4.7
Schematic locations of the measurement planes
90
4.8
The locations of two current segments underneath the
horizontal measurement plane
92
4.9
The node at (0, 0)
96
4.10(a)
The contour plots of a single bounded current segment
with coordinates:
S1=[0.03, 0.07, -0.01; 0.13, 0.07, -0.01]
99
4.10(b)
The contour plots of a single bounded current segment
with coordinates:
S1=[0.13, 0.07, -0.01; 0.03, 0.07, -0.01]
99
4.10(c)
The contour plots of a single bounded current segment
with coordinates:
S1=[0.03, 0.07, -0.05; 0.13, 0.07, -0.01]
100
4.10(d)
The contour plots of a single bounded current segment
with coordinates:
S1=[0.13, 0.07, -0.05; 0.03, 0.07, -0.01]
100
4.10(e)
The contour plots of a single bounded current segment
with coordinates:
S1=[0.10, 0.10, -0.01; 0.03, 0.03, -0.05]
101
xix
4.10(f)
The contour plots of a single bounded current segment
with coordinates:
S1=[0.05, 0.05, -0.01; 0.13, 0.12, -0.05]
101
4.11
The contour plots for two current segments at different
locations as the distance between them decreases
104
4.12(a)
The contour plots for two bounded current segments, S1
and S2 with coordinates;
S1=[0.03, 0.03, -0.05; 0.13, 0.03, -0.01] and
S2=[0.03, 0.13, -0.05; 0.13, 0.13 –0.01]
105
4.12(b)
The contour plots for two bounded current segments, S1
and S2 with coordinates;
S1=[0.02, 0.08, -0.01; 0.12, 0.02, -0.01] and
S2=[0.02, 0.08, -0.01; 0.12, 0.13 –0.01]
106
4.12(c)
The contour plots for two bounded current segments, S1
and S2 with coordinates;
S1=[0.02, 0.08, -0.05; 0.12, 0.02, -0.01] and
S2=[0.02, 0.08, -0.05; 0.12, 0.13 –0.01]
106
4.13
Magnetic fields measurements at locations C, D and E
108
4.14
Magnetic field measured at point M due to two current
sources
109
4.15
The measurement point M located at equal distance
between the current sources
112
4.16
Magnetic field measured at point M due to two current
sources at different distances
113
4.17
Magnetic field measured at point M due to two current
sources at different distances
114
4.18
Magnetic field measured at point M due to two current
sources at different distances
115
4.19
Magnetic field measured at point M due to two current
sources at different distances
116
5.1
The improved version of FTTM2
118
5.2
Fuzzy Topographic Topological Mapping Version 2
(FTTM2)
118
5.3
The transformation of FTTM1 to FTTM2
122
xx
5.4
The linear stretching of magnetic fields data to image
data
124
5.5
The possible ways for a seed pixel to grow using fourconnectivity
128
5.6
The fuzzification graph
129
5.7
The defuzzification graph
131
6.1
The induced partition of A on B
143
6.2
The preservation of partitions from the first component
to the fourth component
145
7.1
Flowchart of the implementation process
148
7.2
The xy-plane where the measurement of B z is made
151
7.3
The xz-plane where the measurement of B y (left ) is made
151
7.4
The xz-plane where the measurement of B y (lright ) is made
152
7.5
The contour plot for the data B z measured on the top of
the head
155
7.6
The plot of B z against y
156
7.7
The plot of the image data against y
159
7.8(a)
The clustered data C1
174
7.8(b)
The clustered data C2
174
7.9(a)
The clustered data C1
175
7.9(b)
The clustered data C2
175
7.10(a)
The clustered data C1
176
xxi
7.10(b)
The clustered data C2
176
7.11(a)
The clustered data C1
177
7.11(b)
The clustered data C2
178
7.11(c)
The clustered data C3
178
7.12(a)
The clustered data C1
179
7.12(b)
The clustered data C2
179
7.12(c)
The clustered data C3
180
7.12(d)
The clustered data C4
180
7.13
The output of the SBRG algorithm
182
7.14(a)
The 3-dimensional surface plot of the defuzzified data
185
7.14(b)
The 3-dimensional surface plot (rotated) of the
defuzzified data depicting the valley between the two
peaks
186
7.15
The plot of max(Z) values against x
189
7.16
The schematic diagram of the current segment below the
measurement plane
189
7.17
The contour plot of the magnetic fields obtained from
measurement at the left side of the head
195
7.18
The plot of data B y (left ) against z
196
7.19
The plot of the image data against y
197
7.20(a)
The clustered data C1
198
7.20(b)
The clustered data C2
199
7.21
The output of the SBRG algorithm
200
xxii
7.22(a)
The surface plot
203
7.22(b)
The surface plot (rotated)
204
7.23
The contour plot as viewed from the right side of the
head
207
7.24
The plot of data B y (right ) against y
207
7.25
The plot of image data against y
209
7.26(a)
The clustered data C1
210
7.26(b)
The clustered data C2
210
7.27
The output of SBRG algorithm
211
7.28
The intersection of the three planes
212
7.29
The 520A fluxgate magnetometer
213
7.30
The fluxgate magnetometer probe (model APS 550 3
Axis)
213
7.31
Schematic diagram of the experimental set-up
215
7.32
Equipment set-up in the laboratory for measuring the
magnetic fields
216
7.33
The contour plot of the magnetic fields obtained through
experiments
221
7.34
The plot of Bn against y
222
7.35
The plot of image data against y
223
7.36
The output of the SBRG algorithm
224
7.37
The surface plot of the defuzzified data
226
7.38
The plot of max(Z) against x
227
xxiii
7.39
The schematic diagram of the current segment below the
measurement plane
228
7.40
The contour plots of magnetic fields on (a) the top of the
head, (b) left side of the head and (c) right side of the
head
232
7.41
The contour plots for the experimental data
234
7.42
The contour plots for the experimental data
235
xxiv
LIST OF SYMBOLS AND ABBREVIATIONS
General symbols
A, B , C ,...
-
Arbitrary set (crisp or fuzzy)
A=B
-
Set equality
A≠ B
-
Set inequality
A⊂ B
-
Proper set inclusion
A⊆ B
-
Set inclusion
A∩ B
-
Set intersection
A∪ B
-
Set union
A\ B
-
Set difference
-
Closed interval of real numbers between a and b
-
Interval of real numbers open in a and closed in b
-
Interval of real numbers closed in a and open in b
-
Open interval of real numbers between a and b
Gmin
-
The smallest possible gray level value
Gmax
-
The largest possible gray level value
{x1 , x2 , , x3 ,...}
-
Set of elements x1 , x 2 , , x3 ,...
{x : p(x )}
-
Set determined by property p
(x1 , x2 , , x3 ,...)
-
n-tuple
i, j, k,…
-
Arbitrary identifiers (indices)
N 4 ( p)
-
The 4-neighbours of p
[x ]
-
Matrix
∈
-
Element of
∉
-
Not an element of
∃
-
There exist (at least one)
[a , b]
(a , b]
[a , b )
(a , b )
ij
xxv
∀
-
For all
ℜ
-
Set of real numbers
ℜn
-
Set of n-tuple of real numbers
U or X
-
Universal set
Ø
-
Empty set
μ A (x )
-
Membership grade of x in fuzzy set A
...
-
Euclidean norm
(D…)
-
Definition
∧
-
Intersection of two fuzzy sets
∨
-
Union of two fuzzy sets
min [x1 , x2 , x3 ,..., xn ]
-
Minimum of x1 , x2 , x3 ,..., xn
max [x1 , x2 , x3 ,..., xn ]
-
Maximum of x1 , x2 , x3 ,..., xn
m
-
The weighting exponent
U
-
Fuzzy partition matrix
V = {v1 ,v2, ,v3 ,...,vc }
-
Set of cluster centres
Magnetism
Bz
- The magnetic fields data measured on the top of the head (at
z = 0)
B y (left )
- The magnetic fields data measured on the left side of the head
(at y = 0 )
B y (right )
- The magnetic fields data measured on the right side of the head
(at y = 0.15 )
μ0
- The permeability in free space
μ
- Material permeability
I
- The magnitude of current flow in a conductor
θ1
- The angle between a line parallel to the segment in the
direction of the current flow and the line joining the starting
point of the segment A to the measurement point
θ2
- The angle between a line parallel to the segment in the
direction of the current flow and the line joining the end point
of the current segment to the measurement point
xxvi
- The perpendicular distance between the current source and the
measurement point on a plane
r
The unit perpendicular distance
r̂
BZ
- The absolute value of B z
max(BZ)
- The maximum value of BZ
min(BZ)
- The maximum value of BZ
max(max(BZ)) - The maximum among all the maximum values of BZ
max(max(BZ)) - The minimum among all the minimum values of BZ
Iz
- The image data in the range of [0, 255]
max( I z )
- The maximum value of the fuzzified data
min( I z )
- The minimum value of the fuzzified data
μI
- The fuzzified image data in the range [0, 1]
z
μB
z
- The fuzzified magnetic fields values in the range [0, 1]
x, y, z
- Cartesian coordinates
Bx
- The x-component of the magnetic fields
By
- The y-component of the magnetic fields
B0
- The magnetic fields data when current is not flowing
Bc
- The magnetic fields data when current is flowing
Bn
- The net magnetic fields data obtained from the difference
between Bc and B0
h
- The distance between the current source and the measurement
plane
d
- The distance between the two extrema of the magnetic field
contours
Z
- The defuzzified values
xxvii
B
- Magnetic fields vector
H
- Magnetic field intensity
âφ
- The unit vector along the concentric circular path of the
magnetic field lines
â r
- The unit vector along the perpendicular line from the line
current to the field point
â I
- The unit vector along the line current
Abbreviations
MEG
- Magnetoencephalography
SQUID
- Superconducting Quantum Interference Device
FTTM1 - Fuzzy Topographic Topological Mapping Version 1
FTTM2 - Fuzzy Topographic Topological Mapping Version 2
FCM
- Fuzzy c-Means
SBRG
- Seed-Based Region Growing
MC
- Magnetic contour plane
BM
- Base magnetic plane
FM
- Fuzzy magnetic field
TM
- Topographic magnetic field
IC
- Image contour plane
BI
- Base image plane
FI
- Fuzzy image field
TI
- Topographic image field
ROI
- Region of interest
S1
- Current segment 1
S2
- Current segment 2
MUSIC - Multiple signal classification
FEM
- Finite element method
BEM
- Boundary element method
ICA
- Independent component analysis
xxviii
ECD
- Equivalent current dipole
SEFs
- Somatically evoked magnetic fields
VEFs
- Visually evoked magnetic fields
PC
- Partition coefficient
PE
- Partition entropy
Prop_E
- Proportion exponent
CSV
- Compactness and separation validity
PSPs
- Postsynaptic potentials
MNE
- Minimum norm estimates
LHe
- Liquid helium
RHR
- Right-Hand Rule
MLE
- Maximum likelihood estimation
xxix
LIST OF APPENDICES
APPENDIX
TITLE
PAGE
A1
Flowchart 1
252
A2
Flowchart 2
253
A3
Flowchart 3
254
A4
Flowchart 4
255
A5
Flowchart 5
256
A6
Flowchart 6
257
A7
Flowchart 7
258
B
Flowchart 8
259
C
Flowchart 9
260
D1
Flowchart 10
261
D2
Flowchart 11
262
D3
Flowchart 12
263
D4
Flowchart 13
264
E
Trends of the results
265
F
Papers Published
272